2003 Academy Colloquium Series

Connections between Mathematics and Physics

Sergei Gukov (CMI/Harvard University)

Abstract

Throughout the history of physics and mathematics, there are many occasions when progress in one subject is motivated by the developments in the other. Starting with a brief review of the history of this phenomenon, I will focus more closely on topological ideas in physics, which led to many exciting discoveries during the last twenty years. In the last part of the talk, I will briefly mention some recent developments, inspired by the ideas in string theory. Lecture notes

Dynamics on the Space of Lattices and Number Theory

Elon Lindenstrauss (CMI/Stanford University)

Abstract

A very active area of current research is dynamics on locally homogeneous spaces. The simplest, and arguably the most important example is $SL_n(R) / SL_n(Z)$, which can be identified with the space of lattices in $R^n$ which have a fundamental domain of volume 1. The study of unipotent flows on $SL_3(R) / SL_3(Z)$ has enabled G.A. Margulis to prove in 1986 a conjecture of Oppenheim stated in 1929 about values of indefinite quadratic forms. Subsequently, M. Ratner proved a general, and extremely powerful result on measures and sets invariant under unipotent flows which has found diverse applications in many fields of mathematics. On the opposite extreme from unipotent flows are Cartan flows which are much less understood and which are the subject of vigorous research. Understanding the action of the group of diagonal matrices on $SL_3(R) / SL_3(Z)$ will already imply a longstanding conjecture of Littlewood. Several authors have given partial results about these actions, and these partial results already have nice applications, for example to proving equidistribution of eigenfunctions of the Laplacian, a problem commonly known as Quantum Unique Ergodicity, in the arithmetic context. I will also discuss the relation between diophantine approximations and dynamics on the space of lattices, which enabled D. Kleinbock and G.A. Margulis to answer some long-standing conjectures in that field, and a recent extension of these results which was obtained jointly with D. Kleinbock and B. Weiss. Lecture notes

Elliptic Functions

Pavel Etingof (MIT)

Abstract

The talk will contain an elementary introduction to elliptic functions and a discussion of their connection with combinatorics and elementary number theory. Lecture notes

The Fractal Geometry of the Mandelbrot Set

Robert L. Devaney (Boston University)

Abstract

In this lecture we describe several folk theorems concerning the Mandelbrot set. While this set is extremely complicated from a geometric point of view, we will show that, as long as you know how to add and how to count, you can understand this geometry completely. We will encounter many famous mathematical objects in the Mandelbrot set, like the Farey tree and the Fibonacci sequence. And we will find many soon-to-be-famous objects as well, like the "Devaney" sequence. There might even be a joke or two in the talk.

The Pythogorean Theorem and the Nine-Point Circle

Roger Howe (Yale University)

Abstract

The Pythagorean Theorem is fundamental to Euclidean geometry, and is probably its best known result. The Nine-Point Circle is a landmark discovery of the renaissance of Euclidean geometry that extended from the 18th to early 20th centuries. This talk will explore the relationship between the two. Lecture notes

The Strong Perfect Graph Theorem

Maria Chudnovsky (CMI/Princeton University)

Abstract

A graph is a set of vertices some pairs of which are joined by an edge and others are not. One of the properties of graphs studied in graph theory is graph coloring: color the vertices with as few colors as possible so that no two vertices with an edge between them receive the same color. Clearly, if a graph contains 10 vertices every two of which are joined by an edge, we need at least 10 colors in order to color it. However even that may not be enough. It has been a central questions in graph theory to characterize all graph for which it would be enough. In 1961 Claude Berge conjectured a possible characterization (that became known as The Strong Perfect Graph Conjecture), in 2002, in join work with Neil Robertson, Paul Seymour and Robin Thomas, we proved it. In my talk I will explain the problem more precisely and discuss some ideas used in the proof. Lecture notes

Are Cubics Rational?

Joe Harris (Harvard University)

Abstract

The geometry of cubic polynomials, and in particular the rationality of cubic hyper surfaces, has been a catalyst for new developments in algebraic geometry for two centuries. The discovery of the irrationality of cubic curves, for example, led to the study of abelian integrals, which was central to much of 19th century mathematics; while the rationality of cubic surfaces in many ways gave birth to the subject of birational geometry. In the 20th century, Clemens' and Griffiths' proof of the irrationality of cubic threefold provided not just the first example of a counterexample to Luroth's theorem in higher dimensions, but a magnificent example of how Hodge theory could be used to settle algebrao-geometric questions. In this talk, I'd like to review these developments and then to focus on the next, and currently unsettled, case: the rationality of cubic four folds. Here evidence obtained by Brendan Hassett and others suggests that cubic fourfolds may also play a pivotal role, providing among other things an answer to the questions of whether rationality is an open or a closed condition in smooth families. I'll discuss the current state of our knowledge, and what is conjectured.